Integrand size = 15, antiderivative size = 153 \[ \int \frac {\cot ^3(x)}{a+b \sin ^3(x)} \, dx=\frac {b^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3}}-\frac {\csc ^2(x)}{2 a}-\frac {\log (\sin (x))}{a}-\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (x)\right )}{3 a^{5/3}}+\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (x)+b^{2/3} \sin ^2(x)\right )}{6 a^{5/3}}+\frac {\log \left (a+b \sin ^3(x)\right )}{3 a} \] Output:
1/3*b^(2/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*sin(x))*3^(1/2)/a^(1/3))*3^(1/2) /a^(5/3)-1/2*csc(x)^2/a-ln(sin(x))/a-1/3*b^(2/3)*ln(a^(1/3)+b^(1/3)*sin(x) )/a^(5/3)+1/6*b^(2/3)*ln(a^(2/3)-a^(1/3)*b^(1/3)*sin(x)+b^(2/3)*sin(x)^2)/ a^(5/3)+1/3*ln(a+b*sin(x)^3)/a
Time = 0.36 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.93 \[ \int \frac {\cot ^3(x)}{a+b \sin ^3(x)} \, dx=\frac {-3 a^{2/3} \csc ^2(x)-6 a^{2/3} \log (\sin (x))+2 \left (a^{2/3}-(-1)^{2/3} b^{2/3}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{a}-\sqrt [3]{b} \sin (x)\right )+2 \left (a^{2/3}-b^{2/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (x)\right )+2 \left (a^{2/3}+\sqrt [3]{-1} b^{2/3}\right ) \log \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (x)\right )}{6 a^{5/3}} \] Input:
Integrate[Cot[x]^3/(a + b*Sin[x]^3),x]
Output:
(-3*a^(2/3)*Csc[x]^2 - 6*a^(2/3)*Log[Sin[x]] + 2*(a^(2/3) - (-1)^(2/3)*b^( 2/3))*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sin[x]] + 2*(a^(2/3) - b^(2/3))* Log[a^(1/3) + b^(1/3)*Sin[x]] + 2*(a^(2/3) + (-1)^(1/3)*b^(2/3))*Log[a^(1/ 3) + (-1)^(2/3)*b^(1/3)*Sin[x]])/(6*a^(5/3))
Time = 0.44 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3042, 3709, 2373, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cot ^3(x)}{a+b \sin ^3(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (x)^3 \left (a+b \sin (x)^3\right )}dx\) |
| \(\Big \downarrow \) 3709 |
| \(\displaystyle \int \frac {\left (1-\sin ^2(x)\right ) \csc ^3(x)}{a+b \sin ^3(x)}d\sin (x)\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \int \left (\frac {b \left (\sin ^2(x)-1\right )}{a \left (a+b \sin ^3(x)\right )}+\frac {\csc ^3(x)}{a}-\frac {\csc (x)}{a}\right )d\sin (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3}}+\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (x)+b^{2/3} \sin ^2(x)\right )}{6 a^{5/3}}-\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (x)\right )}{3 a^{5/3}}+\frac {\log \left (a+b \sin ^3(x)\right )}{3 a}-\frac {\csc ^2(x)}{2 a}-\frac {\log (\sin (x))}{a}\) |
Input:
Int[Cot[x]^3/(a + b*Sin[x]^3),x]
Output:
(b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]* a^(5/3)) - Csc[x]^2/(2*a) - Log[Sin[x]]/a - (b^(2/3)*Log[a^(1/3) + b^(1/3) *Sin[x]])/(3*a^(5/3)) + (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[x] + b^ (2/3)*Sin[x]^2])/(6*a^(5/3)) + Log[a + b*Sin[x]^3]/(3*a)
Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[E xpandIntegrand[(c*x)^m*(Pq/(a + b*x^n)), x], x] /; FreeQ[{a, b, c, m}, x] & & PolyQ[Pq, x] && IntegerQ[n] && !IGtQ[m, 0]
Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + ( f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si mp[ff^(m + 1)/f Subst[Int[x^m*((a + b*(c*ff*x)^n)^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(\frac {2 \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2} a}-i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (27 \textit {\_Z}^{3} a^{5}-27 i a^{4} \textit {\_Z}^{2}-9 \textit {\_Z} \,a^{3}+i a^{2}-i b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-\frac {6 a^{2} \textit {\_R}}{b}+\frac {2 i a}{b}\right ) {\mathrm e}^{i x}-1\right )\right )-\frac {\ln \left ({\mathrm e}^{2 i x}-1\right )}{a}\) | \(110\) |
| default | \(-\frac {1}{2 a \sin \left (x \right )^{2}}-\frac {\ln \left (\sin \left (x \right )\right )}{a}+\frac {\left (-\frac {\ln \left (\sin \left (x \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\ln \left (\sin \left (x \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (x \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (x \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\ln \left (a +b \sin \left (x \right )^{3}\right )}{3 b}\right ) b}{a}\) | \(132\) |
Input:
int(cot(x)^3/(a+b*sin(x)^3),x,method=_RETURNVERBOSE)
Output:
2*exp(2*I*x)/(exp(2*I*x)-1)^2/a-I*sum(_R*ln(exp(2*I*x)+(-6/b*a^2*_R+2*I/b* a)*exp(I*x)-1),_R=RootOf(27*_Z^3*a^5-27*I*a^4*_Z^2-9*_Z*a^3+I*a^2-I*b^2))- 1/a*ln(exp(2*I*x)-1)
Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 874, normalized size of antiderivative = 5.71 \[ \int \frac {\cot ^3(x)}{a+b \sin ^3(x)} \, dx=\text {Too large to display} \] Input:
integrate(cot(x)^3/(a+b*sin(x)^3),x, algorithm="fricas")
Output:
-1/12*(2*(a*cos(x)^2 - a)*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1 /54*(a^2 - b^2)/a^5)^(1/3) - 2/a)*log(-1/2*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)*a^2 - b*sin(x) - a) - ( (a*cos(x)^2 - a)*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a) + 3*sqrt(1/3)*(a*cos(x)^2 - a)*sqrt(-((3*(I*sqrt( 3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)^2*a ^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5 )^(1/3) - 2/a)*a + 4)/a^2) + 6*cos(x)^2 - 6)*log(1/2*(3*(I*sqrt(3) + 1)*(- 1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)*a^2 + 3/2*sqr t(1/3)*a^2*sqrt(-((3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)^2*a^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b ^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)*a + 4)/a^2) - 2*b*sin(x) + a) - ((a*cos(x)^2 - a)*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a ^2 - b^2)/a^5)^(1/3) - 2/a) - 3*sqrt(1/3)*(a*cos(x)^2 - a)*sqrt(-((3*(I*sq rt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)^ 2*a^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/ a^5)^(1/3) - 2/a)*a + 4)/a^2) + 6*cos(x)^2 - 6)*log(-1/2*(3*(I*sqrt(3) + 1 )*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)*a^2 + 3/2 *sqrt(1/3)*a^2*sqrt(-((3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54* (a^2 - b^2)/a^5)^(1/3) - 2/a)^2*a^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/a^3 +...
\[ \int \frac {\cot ^3(x)}{a+b \sin ^3(x)} \, dx=\int \frac {\cot ^{3}{\left (x \right )}}{a + b \sin ^{3}{\left (x \right )}}\, dx \] Input:
integrate(cot(x)**3/(a+b*sin(x)**3),x)
Output:
Integral(cot(x)**3/(a + b*sin(x)**3), x)
Time = 0.11 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.99 \[ \int \frac {\cot ^3(x)}{a+b \sin ^3(x)} \, dx=-\frac {\sqrt {3} {\left (b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} - \frac {2 \, a}{b}\right )} + 2 \, a\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (x\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2}} + \frac {{\left (2 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 1\right )} \log \left (\sin \left (x\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (x\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (\left (\frac {a}{b}\right )^{\frac {2}{3}} - 1\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (x\right )\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (\sin \left (x\right )\right )}{a} - \frac {1}{2 \, a \sin \left (x\right )^{2}} \] Input:
integrate(cot(x)^3/(a+b*sin(x)^3),x, algorithm="maxima")
Output:
-1/9*sqrt(3)*(b*(3*(a/b)^(1/3) - 2*a/b) + 2*a)*arctan(-1/3*sqrt(3)*((a/b)^ (1/3) - 2*sin(x))/(a/b)^(1/3))/a^2 + 1/6*(2*(a/b)^(2/3) + 1)*log(sin(x)^2 - (a/b)^(1/3)*sin(x) + (a/b)^(2/3))/(a*(a/b)^(2/3)) + 1/3*((a/b)^(2/3) - 1 )*log((a/b)^(1/3) + sin(x))/(a*(a/b)^(2/3)) - log(sin(x))/a - 1/2/(a*sin(x )^2)
Time = 0.12 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.94 \[ \int \frac {\cot ^3(x)}{a+b \sin ^3(x)} \, dx=\frac {b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (x\right ) \right |}\right )}{3 \, a^{2}} + \frac {\log \left ({\left | b \sin \left (x\right )^{3} + a \right |}\right )}{3 \, a} - \frac {\log \left ({\left | \sin \left (x\right ) \right |}\right )}{a} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \sin \left (x\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (\sin \left (x\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (x\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2}} - \frac {1}{2 \, a \sin \left (x\right )^{2}} \] Input:
integrate(cot(x)^3/(a+b*sin(x)^3),x, algorithm="giac")
Output:
1/3*b*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(x)))/a^2 + 1/3*log(abs(b*si n(x)^3 + a))/a - log(abs(sin(x)))/a - 1/3*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/ 3*sqrt(3)*((-a/b)^(1/3) + 2*sin(x))/(-a/b)^(1/3))/a^2 - 1/6*(-a*b^2)^(1/3) *log(sin(x)^2 + (-a/b)^(1/3)*sin(x) + (-a/b)^(2/3))/a^2 - 1/2/(a*sin(x)^2)
Time = 38.06 (sec) , antiderivative size = 2003, normalized size of antiderivative = 13.09 \[ \int \frac {\cot ^3(x)}{a+b \sin ^3(x)} \, dx=\text {Too large to display} \] Input:
int(cot(x)^3/(a + b*sin(x)^3),x)
Output:
symsum(log(-(256*(64*b^7*tan(x/2) + 32*a*b^6 - 44*a^3*b^4 + 15*a^5*b^2 - 1 024*root(27*a^5*e^3 - 27*a^4*e^2 + 9*a^3*e - a^2 + b^2, e, k)*b^8*tan(x/2) ^2 - 84*a^2*b^5*tan(x/2) + 26*a^4*b^3*tan(x/2) + 48*a*b^6*tan(x/2)^2 - 16* root(27*a^5*e^3 - 27*a^4*e^2 + 9*a^3*e - a^2 + b^2, e, k)*a^2*b^6 + 328*ro ot(27*a^5*e^3 - 27*a^4*e^2 + 9*a^3*e - a^2 + b^2, e, k)*a^4*b^4 - 165*root (27*a^5*e^3 - 27*a^4*e^2 + 9*a^3*e - a^2 + b^2, e, k)*a^6*b^2 - 70*a^3*b^4 *tan(x/2)^2 + 25*a^5*b^2*tan(x/2)^2 - 48*root(27*a^5*e^3 - 27*a^4*e^2 + 9* a^3*e - a^2 + b^2, e, k)^2*a^3*b^6 - 915*root(27*a^5*e^3 - 27*a^4*e^2 + 9* a^3*e - a^2 + b^2, e, k)^2*a^5*b^4 + 630*root(27*a^5*e^3 - 27*a^4*e^2 + 9* a^3*e - a^2 + b^2, e, k)^2*a^7*b^2 + 873*root(27*a^5*e^3 - 27*a^4*e^2 + 9* a^3*e - a^2 + b^2, e, k)^3*a^6*b^4 - 810*root(27*a^5*e^3 - 27*a^4*e^2 + 9* a^3*e - a^2 + b^2, e, k)^3*a^8*b^2 + 864*root(27*a^5*e^3 - 27*a^4*e^2 + 9* a^3*e - a^2 + b^2, e, k)^4*a^7*b^4 - 405*root(27*a^5*e^3 - 27*a^4*e^2 + 9* a^3*e - a^2 + b^2, e, k)^4*a^9*b^2 - 1296*root(27*a^5*e^3 - 27*a^4*e^2 + 9 *a^3*e - a^2 + b^2, e, k)^5*a^8*b^4 + 1215*root(27*a^5*e^3 - 27*a^4*e^2 + 9*a^3*e - a^2 + b^2, e, k)^5*a^10*b^2 - 608*root(27*a^5*e^3 - 27*a^4*e^2 + 9*a^3*e - a^2 + b^2, e, k)*a*b^7*tan(x/2) - 8880*root(27*a^5*e^3 - 27*a^4 *e^2 + 9*a^3*e - a^2 + b^2, e, k)^2*a^3*b^6*tan(x/2)^2 + 5067*root(27*a^5* e^3 - 27*a^4*e^2 + 9*a^3*e - a^2 + b^2, e, k)^2*a^5*b^4*tan(x/2)^2 + 1050* root(27*a^5*e^3 - 27*a^4*e^2 + 9*a^3*e - a^2 + b^2, e, k)^2*a^7*b^2*tan...
\[ \int \frac {\cot ^3(x)}{a+b \sin ^3(x)} \, dx =\text {Too large to display} \] Input:
int(cot(x)^3/(a+b*sin(x)^3),x)
Output:
(24*cos(x)*sin(x)*a*b + 128*cos(x)*b**2 + 288*int(tan(x/2)/(tan(x/2)**6*a + 3*tan(x/2)**4*a + 8*tan(x/2)**3*b + 3*tan(x/2)**2*a + a),x)*sin(x)**2*a* b**2 + 64*int(1/(tan(x/2)**9*a + 3*tan(x/2)**7*a + 8*tan(x/2)**6*b + 3*tan (x/2)**5*a + tan(x/2)**3*a),x)*sin(x)**2*a*b**2 + 12*int(1/(tan(x/2)**8*a + 3*tan(x/2)**6*a + 8*tan(x/2)**5*b + 3*tan(x/2)**4*a + tan(x/2)**2*a),x)* sin(x)**2*a**2*b + 192*int(1/(tan(x/2)**7*a + 3*tan(x/2)**5*a + 8*tan(x/2) **4*b + 3*tan(x/2)**3*a + tan(x/2)*a),x)*sin(x)**2*a*b**2 + 20*int(1/(tan( x/2)**6*a + 3*tan(x/2)**4*a + 8*tan(x/2)**3*b + 3*tan(x/2)**2*a + a),x)*si n(x)**2*a**2*b + 512*int(1/(tan(x/2)**6*a + 3*tan(x/2)**4*a + 8*tan(x/2)** 3*b + 3*tan(x/2)**2*a + a),x)*sin(x)**2*b**3 - log(tan(x/2)**2 + 1)*sin(x) **2*a**2 - 64*log(tan(x/2)**2 + 1)*sin(x)**2*b**2 + 3*log(tan(x/2)**6*a + 3*tan(x/2)**4*a + 8*tan(x/2)**3*b + 3*tan(x/2)**2*a + a)*sin(x)**2*a**2 - 8*log(tan(x/2))*sin(x)**2*a**2 + 128*log(tan(x/2))*sin(x)**2*b**2 + 2*sin( x)**2*a**2 + 20*sin(x)**2*a*b*x - 64*sin(x)**2*b**2 + 24*sin(x)*a*b - 4*a* *2 + 128*b**2)/(8*sin(x)**2*a**3)